Some uniqueness, multiplicity and complete dynamics for a cooperative model

In this paper, a two-species cooperative model with diffusion and under homogeneous Dirichlet boundary conditions is investigated. It is shown the existence, stability, uniqueness and multiplicity of positive solutions. In particular, we study the global asymptotical stability of the unique positive solution when a∈(λ1(−bθd1+βθd),∞) and α is large. Our method of analysis is based on perturbation technique, the Lyapunov-Schmidt procedure and the bifurcation theory.